New Examples of Irreducible Subfactors of the Hyperfinite II1 Factor with Rational, Non-Integer Index

Abstract

In his thesis, Schou showed that certain infinite-dimensional commuting squares could be used to construct irreducible hyperfinite subfactors. Bisch used this approach to construct a subfactor with index 4.5 that was the first example of an irreducible hyperfinite subfactor with rational, non-integer index. In this dissertation, we construct new examples of irreducible hyperfinite subfactors with rational, non-integer index. We first show that, for every N ≥ 4, if there exists a symmetric commuting square based on an inclusion graph N-star with A_∞-tail, the resulting irreducible hyperfinite subfactor would have a rational, non-integer index. We then explicitly construct such symmetric commuting squares in the case N = 5, 6, 7 and 9. Thus, there exist irreducible hyperfinite subfactors based on N-stars with A_∞-tail for these N, and their indices are 5 + 1/3, 6 + 1/4, 7 + 1/5 and 9 + 1/7.